# Vitali convergence theorem

Let $f_{1},f_{2},\ldots$ be $\mathbf{L}^{p}$-integrable functions on some measure space, for $1\leq p<\infty$.

The sequence $\{f_{n}\}$ converges in $\mathbf{L}^{p}$ to a measurable function $f$ if and and only if

1. i

the sequence $\{f_{n}\}$ converges to $f$ in measure;

2. ii

the functions $\{\lvert f_{n}\rvert^{p}\}$ are uniformly integrable; and

3. iii

for every $\epsilon>0$, there exists a set $E$ of finite measure, such that $\int_{E^{\mathrm{c}}}\lvert f_{n}\rvert^{p}<\epsilon$ for all $n$.

## Remarks

This theorem can be used as a replacement for the more well-known dominated convergence theorem, when a dominating cannot be found for the functions $f_{n}$ to be integrated. (If this theorem is known, the dominated convergence theorem can be derived as a special case.)

In a finite measure space, condition (iii) is trivial. In fact, condition (iii) is the tool used to reduce considerations in the general case to the case of a finite measure space.

In probability , the definition of “uniform integrability” is slightly different from its definition in general measure theory; either definition may be used in the statement of this theorem.

## References

• 1 Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
• 2 Jeffrey S. Rosenthal. A First Look at Rigorous Probability Theory. World Scientific, 2003.
Title Vitali convergence theorem VitaliConvergenceTheorem 2013-03-22 16:17:10 2013-03-22 16:17:10 stevecheng (10074) stevecheng (10074) 9 stevecheng (10074) Theorem msc 28A20 uniform-integrability convergence theorem ModesOfConvergenceOfSequencesOfMeasurableFunctions UniformlyIntegrable DominatedConvergenceTheorem